Understanding exponents

I am helping my kid with exponents. I told her that the exponent tells us how many times we should multiply the base number. While it works with a simple example like 4^6, I am not sure how to explain her why 4^0 =1 and why 27^(1/3) = 3.

It all relies on knowing the laws ##a^{m+n} = a^m a^n## This is a fundamental property for understanding exponents.

Then it is certainly true that ##1+0 = 1##. So if we put those in the exponents, then it must be true that ##4^{1 + 0} = 4^1##. Thus ##4^1 4^0 = 4^1##. Of course, ##4^1 = 4##. Thus we have something like ##4\cdot 4^0 = 4##. So ##4^0## is some number when multiplied by ##4##, it will give ##4## again. We see immediately that ##4^0 = 1##.

For ##27^{1/3}## something similar holds. Of course we know that ##\frac{1}{3}+ \frac{1}{3} + \frac{1}{3} = 1##. So if we put this in the exponents, we get
[tex]27^{\frac{1}{3}+ \frac{1}{3} + \frac{1}{3}} = 27^1 = 27[/tex]
And when using our fundamental property, we see that
[tex]27^{1/3}27^{1/3}27^{1/3} = 27[/tex]
or just
[tex](27^{1/3})^3 = 27[/tex]
So ##27^{1/3}## is the number such that if we cube it, we get ##27##. But by inspection we see that ##3## is such a number since ##3^3 = 27##, so we must have ##27^{1/3} = 3##.